Does the alternating group $A_5$ contain a subgroup isomorphic to $\Bbb Z_{20}$?

Question

What are all the possible orders of elements in the group $A_5$? Does $A_5$ contain a subgroup isomorphic to $\Bbb Z_{20}$? How about $\Bbb Z_{10}$? How about $\Bbb Z_5$? Justify your answers.

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I've found that the possible orders are 1, 2, 3, and 5. I'm having a hard time conceptualizing this.

The identity in the subgroup must map to the identity in $\Bbb Z_{20}$. There must exist a bijective function from the subgroup to $\Bbb Z_{20}$, and this function must satisfy $f(ab)=f(a)f(b)$.

So I must find a function that works for this? I'm confused.

Answer 1

Hint: a subgroup isomorphic to ${\mathbb Z}_n$ would be generated by an element of order $n$.

Answer 2

Another hint: Any element of $S_n$ can be written as a composition of disjoint cycles. Further, the order of that composition will be the least common multiple of the length of each cycle within that composition.

Note that a $k$-cycle is an element of $A_5 \iff k$ is odd and $k \leq 5$.

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Also note that Cayley's theorem guarantees that $\mathbb{Z}_5$ is a subgroup of $S_5$. Would it also be a subgroup of $A_5$?